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<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">The Crowded-field problem</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">High spatial densities of activated molecules can result in a ‘‘crowded
field problem,’’ in which single molecules are not adequately resolved.
The reason for this is the limitation of the PSF models described
above to fit only a single molecule. To help solve this problem, ThunderSTORM
uses a multiple-emitter fitting analysis (MFA) approach similar to
the algorithm described in <cite class="ltx_cite">[<a href="#bib.bib15" title="Simultaneous multiple-emitter fitting for single molecule super-resolution imaging" class="ltx_ref">2</a>]</cite>.
</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Multiple-emitter fitting analysis</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<p class="ltx_p">The multiple-emitter fitting analysis approach uses a PSF defined
by a model</p>
<table id="Sx1.E1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E1.m1" class="ltx_Math" style="vertical-align:-25px" src="mi/mi14.png" width="295" height="58" alt="\mathrm{PSF}_{N}\left(x,y\mid\boldsymbol{\Theta}\right)=\sum_{i=1}^{N}\mathrm{%
PSF}\left(x,y\mid\boldsymbol{\theta}_{i}\right)\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td>
</tr>
</table>
<p class="ltx_p">where <img id="Sx1.SSx1.p1.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi18.png" width="20" height="16" alt="N"> is a number of molecules allowed in the fitting region,
and <img id="Sx1.SSx1.p1.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi20.png" width="134" height="21" alt="\boldsymbol{\Theta}=\left[\boldsymbol{\theta}_{1},\ldots,\boldsymbol{\theta}_{%
N}\right]">
are parameters describing position and shape of the imaged molecules
modeled by the PSF.</p>
</div>
<div id="Sx1.SSx1.p2" class="ltx_para">
<p class="ltx_p">The fitting of multiple-emitter models to the raw data proceeds according
to the following algorithm. First, the algorithm fits <img id="Sx1.SSx1.p2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi29.png" width="45" height="18" alt="\mathrm{PSF}_{1}">
(a single molecule model) with an initial molecular position. The
fitted PSF is subtracted from the raw data and the position of the
maximum intensity value in the residual image is taken as an approximate
position of a second molecule. The fitting is now repeated on the
raw data with <img id="Sx1.SSx1.p2.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi30.png" width="45" height="18" alt="\mathrm{PSF}_{2}"> (a model containing two molecules)
and with the initial positions estimated in the previous steps. The
result of the fit is subtracted from the raw data to find an approximate
position of a third molecule in the residual image. This routine is
repeated until the maximum number of molecules allowed in the fitting
region is reached. Note that initial positions of the molecules are
further adjusted, during the multiple-emitter fitting analysis, by
a ‘‘Push&amp;Pull" process <cite class="ltx_cite">[<a href="#bib.bib15" title="Simultaneous multiple-emitter fitting for single molecule super-resolution imaging" class="ltx_ref">2</a>]</cite>. To find
the optimal number of molecules, statistical tests are required, see
Section <a href="#Sx1.SSx2" title="Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">Model selection</span></a>.</p>
</div>
<div id="Sx1.SSx1.p3" class="ltx_para">
<p class="ltx_p">Users can specify the size of the fitting region, the maximum number
of molecules allowed in one fitting region, the <a href="PSF.html" title="" class="ltx_ref">type of PSF</a>,
and the <a href="Fitting.html" title="" class="ltx_ref">fitting method</a> (least-squares methods
or maximum likelihood estimation). Optionally, users can constrain
the multiple-emitter fitting algorithm such that all fitted molecules
have the same intensity or an intensity in a given range. The background
offset is constrained to the same intensity for all fitted molecules.</p>
</div>
</div>
<div id="Sx1.SSx2" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Model selection</h2>

<div id="Sx1.SSx2.p1" class="ltx_para">
<p class="ltx_p">Because a model with more parameters will always be able to fit the
data at least as well as a model with fewer parameters <cite class="ltx_cite">[<a href="#bib.bib19" title="The Advanced Theory of Statistics" class="ltx_ref">3</a>, <a href="#bib.bib4" title="Data reduction and error analysis for the physical science" class="ltx_ref">1</a>]</cite>,
statistical tests are required to determine whether the more complex
model provides a significantly better fit of the underlying data.
Statistical tests are usually based on pair-wise model comparison.
Here a fit by <img id="Sx1.SSx2.p1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi29.png" width="45" height="18" alt="\mathrm{PSF}_{1}"> is compared with a fit by <img id="Sx1.SSx2.p1.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi30.png" width="45" height="18" alt="\mathrm{PSF}_{2}">,
the better of the two is compared with a fit by <img id="Sx1.SSx2.p1.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi31.png" width="45" height="18" alt="\mathrm{PSF}_{3}">,
etc. Pair-wise comparisons are based on an F-test <cite class="ltx_cite">[<a href="#bib.bib19" title="The Advanced Theory of Statistics" class="ltx_ref">3</a>, <a href="#bib.bib4" title="Data reduction and error analysis for the physical science" class="ltx_ref">1</a>]</cite>
or on a log-likelihood ratio test <cite class="ltx_cite">[<a href="#bib.bib19" title="The Advanced Theory of Statistics" class="ltx_ref">3</a>, <a href="#bib.bib15" title="Simultaneous multiple-emitter fitting for single molecule super-resolution imaging" class="ltx_ref">2</a>]</cite> as
described below.</p>
</div>
<div id="Sx1.SSx2.SSSx1" class="ltx_subsubsection">
<h3 class="ltx_title ltx_title_subsubsection">F-test</h3>

<div id="Sx1.SSx2.SSSx1.p1" class="ltx_para">
<p class="ltx_p">An F-test <cite class="ltx_cite">[<a href="#bib.bib19" title="The Advanced Theory of Statistics" class="ltx_ref">3</a>, <a href="#bib.bib4" title="Data reduction and error analysis for the physical science" class="ltx_ref">1</a>]</cite> arises in the case of
fitting by <a href="Fitting.html" title="" class="ltx_ref">least squares methods</a>, when we need
to compare significance of the fit between two models, where one model
(the null model) is a special case of the other (the alternative model)
for some choice of parameters. The F-test statistic computed from
the data is given by the formula</p>
<table id="Sx1.E2" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E2.m1" class="ltx_Math" style="vertical-align:-19px" src="mi/mi12.png" width="394" height="48" alt="F_{\chi^{2}}\left(\mathcal{D}\right)=\frac{\left(\chi^{2}\left(\boldsymbol{%
\Theta}_{0}\mid\mathcal{D}\right)-\chi^{2}\left(\boldsymbol{\Theta}_{1}\mid%
\mathcal{D}\right)\right)/\left(v_{1}-v_{0}\right)}{\chi^{2}\left(\boldsymbol{%
\Theta}_{1}\mid\mathcal{D}\right)/\left(n-v_{1}\right)}\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(2)</span></td>
</tr>
</table>
<p class="ltx_p">where the sum of squared residuals <img id="Sx1.SSx2.SSSx1.p1.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi25.png" width="84" height="23" alt="\chi^{2}\left(\boldsymbol{\Theta}\mid\mathcal{D}\right)">
computed for a model with parameters <img id="Sx1.SSx2.SSSx1.p1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi21.png" width="20" height="16" alt="\boldsymbol{\Theta}"> is defined
<a href="Fitting.html" title="" class="ltx_ref">here</a>, vectors <img id="Sx1.SSx2.SSSx1.p1.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi22.png" width="28" height="19" alt="\boldsymbol{\Theta}_{0}"> and
<img id="Sx1.SSx2.SSSx1.p1.m4" class="ltx_Math" style="vertical-align:-5px" src="mi/mi23.png" width="28" height="19" alt="\boldsymbol{\Theta}_{1}"> are parameters of the null and alternative
model, respectively, <img id="Sx1.SSx2.SSSx1.p1.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi35.png" width="21" height="14" alt="v_{0}"> and <img id="Sx1.SSx2.SSSx1.p1.m6" class="ltx_Math" style="vertical-align:-5px" src="mi/mi37.png" width="21" height="14" alt="v_{1}"> (where <img id="Sx1.SSx2.SSSx1.p1.m7" class="ltx_Math" style="vertical-align:-5px" src="mi/mi34.png" width="60" height="16" alt="v_{0}&lt;v_{1}">)
represent the number of free parameters of the null and alternative
model, respectively, and <img id="Sx1.SSx2.SSSx1.p1.m8" class="ltx_Math" style="vertical-align:-2px" src="mi/mi32.png" width="52" height="18" alt="n=l^{2}"> is the number of data points within
the <a href="FittingRegion.html" title="" class="ltx_ref">fitting region</a> <img id="Sx1.SSx2.SSSx1.p1.m9" class="ltx_Math" style="vertical-align:-2px" src="mi/mi28.png" width="18" height="16" alt="\mathcal{D}">.</p>
</div>
<div id="Sx1.SSx2.SSSx1.p2" class="ltx_para">
<p class="ltx_p">Assuming the null hypothesis that the alternative model does not provide
a significantly better fit than the null model, the F-test statistics
computed in Equation (<a href="#Sx1.E2" title="(2) ‣ F-test ‣ Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>) has an F-distribution with
<img id="Sx1.SSx2.SSSx1.p2.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi27.png" width="128" height="21" alt="\left(v_{1}-v_{0},n-v_{1}\right)"> degrees of freedom. The null hypothesis
is rejected if <img id="Sx1.SSx2.SSSx1.p2.m2" class="ltx_Math" style="vertical-align:-8px" src="mi/mi15.png" width="63" height="23" alt="F_{\chi^{2}}\left(\mathcal{D}\right)"> computed from
the data is greater than the critical value of the <img id="Sx1.SSx2.SSSx1.p2.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi16.png" width="93" height="21" alt="F_{v_{1}-v_{0},n-v_{1}}">
distribution for a user-specified <img id="Sx1.SSx2.SSSx1.p2.m4" class="ltx_Math" style="vertical-align:-5px" src="mi/mi33.png" width="13" height="15" alt="p">-value.</p>
</div>
</div>
<div id="Sx1.SSx2.SSSx2" class="ltx_subsubsection">
<h3 class="ltx_title ltx_title_subsubsection">Log-likelihood ratio test</h3>

<div id="Sx1.SSx2.SSSx2.p1" class="ltx_para">
<p class="ltx_p">To compare between the fits of two models, in the case of fitting
by the <a href="Fitting.html" title="" class="ltx_ref">maximum likelihood estimation method</a>,
we use a model selection criteria based on a log-likelihood ratio
test <cite class="ltx_cite">[<a href="#bib.bib19" title="The Advanced Theory of Statistics" class="ltx_ref">3</a>, <a href="#bib.bib15" title="Simultaneous multiple-emitter fitting for single molecule super-resolution imaging" class="ltx_ref">2</a>]</cite><em class="ltx_emph">.</em> Assuming that one model
(the null model) is a special case of the other (the alternative model)
for some choice of parameters, the log-likelihood ratio is given by
the formula</p>
<table id="Sx1.E3" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E3.m1" class="ltx_Math" style="vertical-align:-19px" src="mi/mi13.png" width="228" height="46" alt="\Lambda\left(\mathcal{D}\right)=-2\ln\left[\frac{L\left(\boldsymbol{\Theta}_{0%
}\mid\mathcal{D}\right)}{L\left(\boldsymbol{\Theta}_{1}\mid\mathcal{D}\right)}%
\right]\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(3)</span></td>
</tr>
</table>
<p class="ltx_p">where the likelihood <img id="Sx1.SSx2.SSSx2.p1.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi17.png" width="77" height="21" alt="L\left(\boldsymbol{\Theta}\mid\mathcal{D}\right)">
of parameters <img id="Sx1.SSx2.SSSx2.p1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi21.png" width="20" height="16" alt="\boldsymbol{\Theta}"> is defined <a href="Fitting.html" title="" class="ltx_ref">here</a>,
<img id="Sx1.SSx2.SSSx2.p1.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi22.png" width="28" height="19" alt="\boldsymbol{\Theta}_{0}"> and <img id="Sx1.SSx2.SSSx2.p1.m4" class="ltx_Math" style="vertical-align:-5px" src="mi/mi23.png" width="28" height="19" alt="\boldsymbol{\Theta}_{1}"> are the
parameters of the null and alternative model, respectively, and <img id="Sx1.SSx2.SSSx2.p1.m5" class="ltx_Math" style="vertical-align:-2px" src="mi/mi28.png" width="18" height="16" alt="\mathcal{D}">
is a <a href="FittingRegion.html" title="" class="ltx_ref">fitting region</a>.
</p>
</div>
<div id="Sx1.SSx2.SSSx2.p2" class="ltx_para">
<p class="ltx_p">The probability distribution of the log-likelihood ratio computed
in Equation (<a href="#Sx1.E3" title="(3) ‣ Log-likelihood ratio test ‣ Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a>), assuming the null hypothesis that
the alternative model does not provide a significantly better fit
than the null model, can be approximated by the <img id="Sx1.SSx2.SSSx2.p2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi24.png" width="23" height="22" alt="\chi^{2}"> distribution
with <img id="Sx1.SSx2.SSSx2.p2.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi36.png" width="58" height="16" alt="v_{1}-v_{0}"> degrees of freedom. This approximation is usually
valid even for small sample sizes <cite class="ltx_cite">[<a href="#bib.bib19" title="The Advanced Theory of Statistics" class="ltx_ref">3</a>]</cite>. Here <img id="Sx1.SSx2.SSSx2.p2.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi35.png" width="21" height="14" alt="v_{0}">
and <img id="Sx1.SSx2.SSSx2.p2.m4" class="ltx_Math" style="vertical-align:-5px" src="mi/mi37.png" width="21" height="14" alt="v_{1}"> (where <img id="Sx1.SSx2.SSSx2.p2.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi34.png" width="60" height="16" alt="v_{0}&lt;v_{1}">) represent the number of free parameters
of the null and alternative models, respectively. The null hypothesis
is rejected if the log-likelihood ratio <img id="Sx1.SSx2.SSSx2.p2.m6" class="ltx_Math" style="vertical-align:-6px" src="mi/mi19.png" width="47" height="21" alt="\Lambda\left(\mathcal{D}\right)">
computed from the data is greater than the critical value of the <img id="Sx1.SSx2.SSSx2.p2.m7" class="ltx_Math" style="vertical-align:-8px" src="mi/mi26.png" width="55" height="24" alt="\chi_{v_{1}-v_{0}}^{2}">
distribution for some <img id="Sx1.SSx2.SSSx2.p2.m8" class="ltx_Math" style="vertical-align:-5px" src="mi/mi33.png" width="13" height="15" alt="p">-value specified by the users.</p>
</div>
</div>
</div>
<div id="Sx1.SSx3" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Guidelines for the choice of parameters</h2>

<div id="Sx1.SSx3.p1" class="ltx_para">
<p class="ltx_p">Multiple emitter fitting analysis (MFA) can be used in high-density
data to estimate the number of molecules detected as a single blob.
We recommend setting 3 to 5 molecules per fitting region. The stability
of the algorithm is improved if the molecular brightness is limited
to a realistic range (perhaps 300 to 5000 photons), depending on the
sample. This range can be estimated by ThunderSTORM by processing
a few frames of the data and evaluating the number of detected photons
from each molecule using the histogram function (this relies on correct
entry of the camera parameters). The algorithm can also be more stable
by forcing all molecules to have the same intensity. Unfortunately,
multiple emitter analysis is a computationally costly method, and
may run quite slowly depending on the user-specified maximum number
of molecules within the fitting area. Note that the size of the fitting
radius might need to be increased slightly to accommodate larger blobs.</p>
</div>
</div>
<div id="Sx1.SSx4" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx4.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="Estimators.html" title="" class="ltx_ref">Sub-pixel 2D localization of molecules</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="PSF.html" title="" class="ltx_ref">Point-spread function (PSF)</a></p>
</div>
</li>
<li id="I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i3.p1" class="ltx_para">
<p class="ltx_p"><a href="Fitting.html" title="" class="ltx_ref">Fitting point-spread function models</a></p>
</div>
</li>
<li id="I1.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i4.p1" class="ltx_para">
<p class="ltx_p"><a href="FittingRegion.html" title="" class="ltx_ref">Definition of the fitting region</a></p>
</div>
</li>
<li id="I1.i5" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i5.p1" class="ltx_para">
<p class="ltx_p"><a href="LocalizationUncertainty.html" title="" class="ltx_ref">Localization uncertainty</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib4" class="ltx_bibitem ltx_bib_book">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">P. R. Bevington and D. K. Robinson</span><span class="ltx_text ltx_bib_year">(2003)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Data reduction and error analysis for the physical science</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_series">McGraw-Hill Higher Education</span>,  <span class="ltx_text ltx_bib_publisher">McGraw-Hill</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><span class="ltx_text isbn ltx_bib_external">ISBN 9780072472271</span></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx2.SSSx1.p1" title="F-test ‣ Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">F-test</span></a>,
<a href="#Sx1.SSx2.p1" title="Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">Model selection</span></a>.
</span>
</li>
<li id="bib.bib15" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[2]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">F. Huang, S. L. Schwartz, J. M. Byars and K. A. Lidke</span><span class="ltx_text ltx_bib_year">(2011)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Simultaneous multiple-emitter fitting for single molecule super-resolution imaging</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Biomedical Optics Express</span> <span class="ltx_text ltx_bib_volume">2</span> (<span class="ltx_text ltx_bib_number">5</span>), <span class="ltx_text ltx_bib_pages"> pp. 1377–93</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1364/BOE.2.001377" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx1.p2" title="Multiple-emitter fitting analysis ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">Multiple-emitter fitting analysis</span></a>,
<a href="#Sx1.SSx2.SSSx2.p1" title="Log-likelihood ratio test ‣ Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">Log-likelihood ratio test</span></a>,
<a href="#Sx1.SSx2.p1" title="Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">Model selection</span></a>,
<a href="#Sx1.p1" title="The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">The Crowded-field problem</span></a>.
</span>
</li>
<li id="bib.bib19" class="ltx_bibitem ltx_bib_book">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[3]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">M. Kendall and A. Stuart</span><span class="ltx_text ltx_bib_year">(1979)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">The Advanced Theory of Statistics</span>,
</span>
<span class="ltx_bibblock"> <span class="ltx_text ltx_bib_publisher">London: Charles Griffin</span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx2.SSSx1.p1" title="F-test ‣ Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">F-test</span></a>,
<a href="#Sx1.SSx2.SSSx2.p1" title="Log-likelihood ratio test ‣ Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">Log-likelihood ratio test</span></a>,
<a href="#Sx1.SSx2.SSSx2.p2" title="Log-likelihood ratio test ‣ Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">Log-likelihood ratio test</span></a>,
<a href="#Sx1.SSx2.p1" title="Model selection ‣ The Crowded-field problem" class="ltx_ref"><span class="ltx_text ltx_ref_title">Model selection</span></a>.
</span>
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